Ciucu–Krattenthaler enumeration conjecture for stationary-state coefficients

Write $p$ repeated opening parentheses as $(^p$ and $p$ repeated opening parentheses followed by $p$ closing parentheses as $(\ldots)_p=(^p\ldots)^p$. For matchings of the form (_s_t)_p, write $[s,t,p]$. Let $a_F$ be the coefficient of matching $F$ in the stationary state of $H^{\rm C}_{2p+2s+2t}$. Ciucu–Krattenthaler stationary-state coefficient conjecture. For $F=[s,t,p]$,

$$a_{[s,t,p]}=\det_{1\leq i,j\leq s}\left(\binom{2(s+t+p)+j-2i}{s+t-j}-\binom{2(s+t+p)+j-2i}{s+t-j-2i+1}\right)$$

and equivalently

$$a_{[s,t,p]}=\prod_{j=1}^{s}\frac{(j-1)!(2t+2p+2j-1)!(2p+2j)_j(3t+2p+3j)_{s-j}}{(t+2p+s+2j-1)!(t+s-j)!}.$$

The claim is attributed to Mitra et al. and uses enumerations of hexagons with cut-off corners by Ciucu and Krattenthaler. Its resolution is not stated in the supplied text.